The Pythagorean or Just (meaning true) major tone is the frequency ratio of 9:8, which is 204 cents.

This is easily calculated by subtracting the Harmonic Logarithm of 8 from that of 9 (see page 4 of PDF, centre table).

If we take the outputs of two tone-generators, nominally a whole-tone (200 cents) apart, divide the higher by 9 and the lower by 8, then the result will be two voices separated by a nominal 4 cents. We will have extracted the difference between the Just and the Equal-Temperament whole-tone.

The beauty of this is that digital instruments are already awash with real or implied divide-by-8s; we don't need to add any more. We just need to add the modulo-9 divider.
(Don't fret, that's just a fanciful name for divide-by-9, and is dead easy to implement using a single CD4017 or HC163 )

Another plus is that the output from a modulo-9 divider is not naturally a square-wave and will therefore contain even harmonics.
The harmonic structure does not conform to that of the boring old staircase/ramp and so, even though it has joined the chorus, it will stand out from the crowd.

Four cent singers.jpg

Description:

Fig.1: Simple implementation with duty-cycle of 55.5%

Filesize:

55.96 KB

Viewed:

2639 Time(s)

Last edited by Tony Deff on Fri Feb 08, 2013 9:39 am; edited 1 time in total

If the link from pin 11 (Y9) to pin 15 (reset) is made with a resistor, then grounding pin 15 with a switch will revert the 4017 to its native divide-by-10 mode (because the reset is active-high).

By cascading two such ICs and selecting one of the two reset pins by a two-way switch to ground [Fig. (a)],
then the final output will switch between 50% (divide-by-10 mode) and 55.5% (divide-by-9 mode)†.
This will allow a rapid audio comparison of the tonal difference, the constant divide-by-90 avoiding any confusion caused by a change of pitch.

Furthermore, if the clock input is taken from pin 6 of a 74HC4060 running at 5.0688MHz, you will have a handy A4 = 440Hz reference ...

5.0688 MHz. = 440.0 Hz.
9 × 10 × 128

For a 110Hz reference (guitar 5th string), use 'HC4060 output pin 13 instead, and generate a 33.3% duty-cycle.
You can also tune 220Hz and 440Hz to this because of the 2nd and 4th harmonics. The absence of 3rd, 6th and 9th harmonics reduces confusion (see tables below).

† Pin 12 ("Out") is active-high for the sum of time-slots Y0+Y1+Y2+Y3+Y4.
If the count is aborted at time-slot Y9 (via the reset link) then the active-high width of Out is unaffected, but the duty-cycle is changed.

In [Fig. (b)], the pulse width can be extended by adding one or two further time-slots to the 5 already summed by pin 12.
Q1 is a chopper/mixer for the analogue envelope, which will invert the pulse, so that the rectangular pulse width is
44.4% (S1 & S2 open)
33.3% (S1 closed)
22.2% (S1 & S2 closed)

There are 4 theoretical tone-colours to choose from when using a divided-by-9 rectangular wave.
In practice, the 11% (1/9) ratio is too harsh to be of much conventional use (too many intense harmonics).

The first table calculates absolute voltage levels for a 1V amplitude pulse, disregarding that the more a pulse deviates from square, the weaker its fundamental — its power is taken by the up-coming new harmonics. Because of this, you are liable to whack-up the volume to compensate, which will also boost the harmonics. Consequently, the second table computes the harmonic levels as relative to the fundamental, which suddenly feel a lot more significant. This approach also throws-up the strange fact that the relative powers of the 8th and 10th harmonics are constant, regardless of duty-cycle.

Of course, you could always build your own waveform, using 8 variable resistors (Fig C), summing from any 8 of outputs Y0 - Y8 . The unused output serves a a zero reference. The harmonic structure would be anybody's guess!

It would be wise to avoid deep chasms in the pyramids (Fig D), as the screeching high harmonics — produced when the workers fall in — produces much the same result as the 11% pulse.

It's best understood if you've wasted endless late nights of your life trying to understand implausible TV thrillers.
My presentation is based on the standard modus operandii of several episodes linked by flash-backs and confusing
sub-plots, based on an intial implausible premise of hot air cloaked by smoke and mirrors and further obfuscated
and lent undue gravitas by quasi-scientific mumbo-jumbo tech-speak (given mystic importance here by hyperlinks).

Having invested so much time trying to understand it, you are forced to view to the end in case you miss the vital bit
that makes sense of it all. I don't want to give the plot away so that you then stop viewing, but since you are the only
one to admit to giving-up in confusion, I will admit to being an ivory-tower-based virtual constructor, frustrated in the
real (surreal?) world by long-term illness. As with the TV programs on mind control, where the subject later carries-out
unconscious actions, I will be content if (even if many years from now) someone has an inventive stroke of genius kick-
started by something filtered subconsciously from the sticky goo of data posted here.

If you understood that, then either the smoke canister's empty or my mirror's broke.

You cannot post new topics in this forumYou cannot reply to topics in this forumYou cannot edit your posts in this forumYou cannot delete your posts in this forumYou cannot vote in polls in this forumYou cannot attach files in this forumYou can download files in this forum

Please support our site. If you click through and buy from our affiliate partners, we earn a small commission.